Blood flow in arteries is characterized by pulse pressure waves due to the interaction with the vessel walls. A 3D fluid–structure interaction (FSI) model in a compliant vessel is used to represent the pressure wave propagation. The 3D fluid is described through a shear-thinning generalized Newtonian model and the structure by a nonlinear hyperelastic model. In order to cope with the spurious reflections due to the truncation of the computational domain, several absorbing boundary conditions are analyzed. First, a 1D hyperbolic model that effectively captures the wave propagation nature of blood flow in arteries is coupled with the 3D FSI model. Extending previous results, an energy estimate is derived for the 3D FSI–1D coupling in the case of generalized Newtonian models. Secondly, absorbing boundary conditions obtained from the 1D model are imposed directly on the outflow sections of the 3D FSI model, and numerical results comparing the different absorbing conditions in an idealized vessel are presented. Results in a human carotid bifurcation reconstructed from medical images are also provided in order to show that the proposed methodology can be applied to anatomically realistic geometries.